Trivial extensions of gentle algebras and Brauer graph algebras

By Sibylle Schroll. 2014

98 citations recorded by the source corpus.

Abstract We show that two well-studied classes of tame algebras coincide: namely, the class of symmetric special biserial algebras coincides with the class of Brauer graph algebras. We then explore the connection between gentle algebras and symmetric special biserial algebras by explicitly determining the trivial extension of a gentle algebra by its minimal injective co-generator. This is a symmetric special biserial algebra and hence a Brauer graph algebra of which we explicitly give the Brauer graph. We further show that a Brauer graph algebra gives rise, via admissible cuts, to many gentle algebras and that the trivial extension of a gentle algebra obtained via an admissible cut is the original Brauer graph algebra. As a consequence we prove that the trivial extension of a Jacobian algebra of an ideal triangulation of a Riemann surface with marked points in the boundary is isomorphic to the Brauer graph algebra with Brauer graph given by the arcs of the triangulation.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/db33ab3e195360e45535f075279cd527b236b34f

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Citations in the corpus

A Geometric Model for the Module Category of a Gentle Algebra (2018, unknown): “Finally, (G4) holds by deﬁnition. Conversely, we can associate to each gentle algebra A a certain graph called marked Brauer graph ΓA. We refer the reader to [36, Deﬁnition 1.8] for details (see also [40]). This graph can be embedded in a surface SA with boundary (cf. [32, 36]), in such a way that t”
A reduction theorem for τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}-rigid modules (2016, result): “By the main result of [29], these correspond exactly to the Brauer graph algebras.”
A geometric model for the derived category of gentle algebras (2018, unknown): “LE SCHROLL Abstract. In this paper we construct a geometric model for the bounded derived category of a gentle algebra. The construction is based on the ribbon graph associated to a gentle algebra in [60], and is linked to partially wrapped Fukaya categories by the work of [46] and to derived catego”
A complete derived invariant for gentle algebras via winding numbers and Arf invariants (2019, background): “Recenlty, gentle algebras have been associated to triangulations or dissections of surfaces, in connection with cluster algebras [7], with ribbon graphs [43, 44] or with Fukaya categories of surfaces [28].”